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ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

JOSÉ F. FERNANDO

Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

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and

J. M. GAMBOA

Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

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https://doi.org/10.1142/S0129167X12500310Cited by:11 (Source: Crossref)

Abstract

In this work we define a semialgebraic set S ⊂ ℝⁿ to be irreducible if the noetherian ring of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring : Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

Keywords:

AMSC: Primary 14P10, Primary 14P20, Secondary 58A07

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